The Friendship Paradox, which our interventions are based on, is colloquially stated as “your friends have more friends than you” [1]}, [2]}.The phenomenon has also been generalized to the idea that individual attributes and degree are correlated [3]}, e.g. an individual's co-authors are more likely to be cited [4]}, or that friends more active on social media[5]}. The intuition for why the friendship paradox helps obtain well-connected nodes is this: there are few well-connected hubs in real networks, and since they are connected to many other nodes (by definition), obtaining a friend (or neighbor) of a random node is likely to result in a hub with greater likelihood, compared to the case of randomly selecting nodes.
We establish that the friendship paradox is actually not just one statement, but a set of distinct claims (All theorems and proofs are in Supplement §). First, we find an impossibility, i.e. the individual-level friendship paradox cannot hold for all individuals in a network (Theorem REF ). In practice for real networks, it can hold for a large proportion of nodes in the network (Figs. REF  and REF  in Supplement §). Second, we demonstrate that in contrast to this individual view, the average number of friends of friends across the network can be characterized in two ways using the local and global mean defined below. Third, we find that both local and global means are greater than the mean degree of the network, and these means are related through a novel network characteristic we call inversity.
